Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: $ x^{3} + 3 x + 147 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 83 + 38\cdot 149 + 70\cdot 149^{2} + 87\cdot 149^{3} + 66\cdot 149^{4} + 119\cdot 149^{5} + 148\cdot 149^{6} + 122\cdot 149^{7} + 28\cdot 149^{8} + 101\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 88 + 24\cdot 149 + 91\cdot 149^{2} + 134\cdot 149^{3} + 27\cdot 149^{4} + 74\cdot 149^{5} + 59\cdot 149^{6} + 93\cdot 149^{7} + 121\cdot 149^{8} + 25\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 99 + 101\cdot 149 + 122\cdot 149^{2} + 41\cdot 149^{3} + 123\cdot 149^{4} + 86\cdot 149^{5} + 108\cdot 149^{6} + 42\cdot 149^{7} + 56\cdot 149^{8} + 127\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a^{2} + 97 a + 93 + \left(77 a^{2} + 141 a + 8\right)\cdot 149 + \left(81 a^{2} + 83 a + 24\right)\cdot 149^{2} + \left(109 a^{2} + 87 a + 64\right)\cdot 149^{3} + \left(88 a^{2} + 132 a + 26\right)\cdot 149^{4} + \left(69 a^{2} + 25 a + 7\right)\cdot 149^{5} + \left(96 a^{2} + 111 a + 2\right)\cdot 149^{6} + \left(99 a^{2} + 95 a + 125\right)\cdot 149^{7} + \left(19 a^{2} + 91 a + 116\right)\cdot 149^{8} + \left(84 a^{2} + 40 a + 59\right)\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a^{2} + 13 a + 111 + \left(57 a^{2} + 53 a + 117\right)\cdot 149 + \left(8 a^{2} + 73 a + 26\right)\cdot 149^{2} + \left(98 a^{2} + 20 a + 41\right)\cdot 149^{3} + \left(144 a^{2} + 72 a + 138\right)\cdot 149^{4} + \left(108 a^{2} + 54 a + 85\right)\cdot 149^{5} + \left(96 a^{2} + 120 a + 2\right)\cdot 149^{6} + \left(23 a^{2} + 57 a + 122\right)\cdot 149^{7} + \left(140 a^{2} + 99 a + 59\right)\cdot 149^{8} + \left(122 a^{2} + 105 a + 137\right)\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 a^{2} + 47 a + 3 + \left(110 a^{2} + 73 a + 113\right)\cdot 149 + \left(68 a^{2} + 48 a + 32\right)\cdot 149^{2} + \left(48 a^{2} + 97 a + 114\right)\cdot 149^{3} + \left(42 a^{2} + 131 a + 63\right)\cdot 149^{4} + \left(56 a^{2} + 27 a + 101\right)\cdot 149^{5} + \left(89 a^{2} + 103 a + 15\right)\cdot 149^{6} + \left(34 a^{2} + 71 a + 57\right)\cdot 149^{7} + \left(102 a^{2} + 117 a + 8\right)\cdot 149^{8} + \left(31 a^{2} + 126 a + 87\right)\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 a^{2} + 53 a + 19 + \left(40 a^{2} + 42 a + 122\right)\cdot 149 + \left(143 a^{2} + 84 a + 32\right)\cdot 149^{2} + \left(29 a^{2} + 126 a + 77\right)\cdot 149^{3} + \left(71 a^{2} + 143 a + 121\right)\cdot 149^{4} + \left(107 a^{2} + 105 a + 54\right)\cdot 149^{5} + \left(109 a^{2} + 74 a + 56\right)\cdot 149^{6} + \left(53 a^{2} + 28 a + 95\right)\cdot 149^{7} + \left(124 a^{2} + 18 a + 52\right)\cdot 149^{8} + \left(76 a^{2} + 5 a + 28\right)\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 102 a^{2} + 39 a + 110 + \left(14 a^{2} + 103 a + 32\right)\cdot 149 + \left(59 a^{2} + 140 a + 128\right)\cdot 149^{2} + \left(90 a^{2} + 40 a + 25\right)\cdot 149^{3} + \left(64 a^{2} + 93 a + 127\right)\cdot 149^{4} + \left(119 a^{2} + 68 a + 106\right)\cdot 149^{5} + \left(104 a^{2} + 66 a + 18\right)\cdot 149^{6} + \left(25 a^{2} + 144 a + 126\right)\cdot 149^{7} + \left(138 a^{2} + 106 a + 55\right)\cdot 149^{8} + \left(90 a^{2} + 2 a + 73\right)\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 142 a^{2} + 49 a + 139 + \left(146 a^{2} + 33 a + 36\right)\cdot 149 + \left(85 a^{2} + 16 a + 67\right)\cdot 149^{2} + \left(70 a^{2} + 74 a + 9\right)\cdot 149^{3} + \left(35 a^{2} + 22 a + 50\right)\cdot 149^{4} + \left(134 a^{2} + 15 a + 108\right)\cdot 149^{5} + \left(98 a^{2} + 120 a + 34\right)\cdot 149^{6} + \left(60 a^{2} + 48 a + 109\right)\cdot 149^{7} + \left(71 a^{2} + 13 a + 95\right)\cdot 149^{8} + \left(40 a^{2} + 17 a + 104\right)\cdot 149^{9} +O\left(149^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,5,8)(6,9,7)$ |
| $(1,5,7)(2,8,9)(3,4,6)$ |
| $(1,2)(4,5)(6,7)$ |
| $(1,2,3)(4,5,8)(6,7,9)$ |
| $(4,6)(5,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)(4,8)(7,9)$ | $0$ |
| $9$ | $2$ | $(4,6)(5,7)(8,9)$ | $-2$ |
| $9$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,8)(6,7,9)$ | $-3$ |
| $6$ | $3$ | $(1,7,8)(2,9,4)(3,6,5)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(4,5,8)$ | $0$ |
| $12$ | $3$ | $(1,5,7)(2,8,9)(3,4,6)$ | $0$ |
| $18$ | $6$ | $(1,4,7,2,8,9)(3,5,6)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(4,7,8,6,5,9)$ | $1$ |
| $18$ | $6$ | $(1,4,3,5,2,8)(6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
